302 5 Solid State Physics
5.9 The lattice of the NaCl structure which is face-centered is represented in
Fig. 5.3. The shortest inter ionic distance is represented byr. A given sodium
ion is surrounded by 6Cl−ions at a distancer, 12Na+ions at a distance
r√
2 , 8Cl−ions at a distancer√
3 ,6Na+ions at a distancer√
4 , 24Cl−
ions at a distancer√
5, etc. The coulomb energy of of this ion in the field of
all other ions is thereforeEc=−e^2
r(
6
√
1
−
12
√
2
+
8
√
3
−
6
√
4
+
24
√
5
−···
)
whereeis the charge per ion. Series of this sort which consists of pure numbers
depends on the crystal structure and is known as the Madelung constant.Fig. 5.3The lattice of the
NaCl structure
5.10 Er=−
a
r+
b
r^7(a) For equilibrium,Ermust be minimum, so thatdE
dr= 0
a
r^2−
7 b
r^8= 0
Or,r=ro=(7b/a)^1 /^6(b) Energy of attraction,EA=−a
ro
Energy of repulsion,ER=
b
r 07
|EA|
|ER|=
a
bro^6 =a
b·
7 b
a= 7
Force,F=−dV
dr
Attractive forceFA=a
ro^2(r=ro)Repulsive forceFR=− 7 b
ro^8=
− 7 b
ro^2·
a
7 b=−
a
ro^2
Thus the two forces are equal in magnitude.